2.2 Reflections & Transformations

We need to have a firm grasp on our basic graphs, so we can then easily transform them using some simple rules.

6 Basic Graphs:

(1) Identity (2) Squaring( )f x x (3) Cubing

(4) Square Root

(5) Absolute Value

(6) Reciprocal

y x

2( )f x x

2y x

3( )f x x

( )f x x ( )f x x1

( )f xx

3y x

y x y x

1y

x

3 ways to change up a graph – Reflections, Translations, & DilationStart with y = f (x)

Reflections

y = – f (x) y = f (– x) y = f –1 (x)

Reflection in x-axis Reflection in y-axis Reflection in y = x(inverse!)

Translations

Horizontal Shift Vertical Shift

(since it is –c, think the opposite) (x – c) c to the right (x + c) c to the left

take it for what it is +d d up –d d down

shift y = f (x – c) + d

Dilations

a > 1 0 < a < 1

Taller or skinnier(away from x-axis)

Wider or fatter(towards x-axis)

stretch or compress y = af (x)

Let’s Explore! iPad app Desmos (can also access online @ desmos.com )

Start Desmos Find button in top left cornerChoose transformations Reflections of a functionExplore this graph!

touch next to the 5 & delete up to =enter your ownchange box 4 to x = f (y) (use ABC)touching next to the next to the formula will “turn on” each graphAre they what you expected?

x

Launch website

Now let’s explore translationsGo to , under Transformations Translating Any Function delete function in box 1go to put x in ( )

Play with h & k put in negative values/ positive valuesuse slider or enter a # in boxes 5 & 7 or change values in equation directly in box 3

Launch website

Time for DilationsGo to , under Transformations Scaling Any Function delete function in box 1type in x2

Play with a make bigger or smaller – use slider or type in #s

Bring it all together!Describe (yes, write a sentence!) how the basic graph of y = x3 has been changedThen check on Desmos or calculatorTouch & Type in the equation & confirm your description!

y = 2 (x + 3)3 – 2

Launch website

Graph

Homework

#202 Pg 69 #1–43 odd, 38, 40, 42, 44